Absolute value is a mathematician's way of judging numbers by their magnitude rather than their positive/negative value. It is the distance of the number from 0 on a number line.
Let \(x=70\) be one root of the equation \(2\lvert x-10\rvert=x+a,\) where \(a\) is a constant. Then what is the other root?
Let \(S\) be the solution set of the equation \(\lvert 2x-1 \rvert + 3\lvert x-29\rvert=5x-88.\) Then what is \(S?\)
How many ordered pairs \((x, y)\) are there such that \(x\) is a nonzero integer that satisfies \(-8\leq x \leq 8\), \(y\) is a nonzero integer that satisfies \(-2\leq y \leq 2\), and \[\lvert x+y \rvert = \lvert x \rvert + \lvert y \rvert?\]
Details and assumptions
For an ordered pair of integers \((a,b)\), the order of the integers matter. The ordered pair \((1, 2)\) is different from the ordered pair \((2,1) \).
How many ordered triples of integers \( (a, b, c) \) subject to \( -20 < a < b < c < 20,\) are there, such that
\[ \left| a + b + c \right | = |a| + |b| + |c|? \]
If \(20 < a \leq 21\), what is the value of \[ \bigg| 2-a- \big| 6- \lvert a-20 \rvert \big| \bigg| ?\]